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Cell["\<\
This code computes the main experiments in the text: the three-period example \
with exogenous and endogenous information, the payoff from an individual \
deviation in the competitive economy, and mixed-strategy equilibria with \
various values of risk aversion. The 4-period experiment is in a separate \
code.\
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In what follows, many probabilities are extremely close to zero, and \
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(relative) tolerance criterion when integrals are so close to zero. We turn \
off these warnings, but we check that we always get proper numerical answers \
and that all calls for a nonlinear solver converge to an actual solution of \
the desired equation. The interesting part of the analysis happens away from \
the boundaries where probabilities are trifling.\
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Cell["\<\
We assume that all realizations are far away from winsorizing with very large \
probability. Nonetheless, since we are solving this numerically, we can \
explicitly account for winsorizing. We do so as follows: there is an \
underlying realization for spending and the signal that is normal with full \
support, but the actual spending is winsorized at the value at which P_3=pmax.\
\>", "Text",
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Cell["Prior standard deviation of spending (and precision)", "Text",
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Cell["\<\
tbar is the upper bound for taxes; assume that F regime prevails when upper \
bound is insufficient to guarantee p3=pstar, that is for g3>p3star. Also, the \
distribution of g3 is winsorized as discussed above so as to ensure that \
p3<=p3max. \[Sigma] is the variance of the prior. We use tbar and \[Sigma] \
jointly to target the probability that the price level is 2% above target \
(above 1),  and 6% (above target) in period 3; these probabilities are set to \
10% and 5%, respectively\
\>", "Text",
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For future computations, it is convenient to interpolate the second-period \
price (actually, I will interpolate the inverse)\
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The computation below exploits the law of iterated expectations: p2inv under \
no information is the conditional expectation of p3inv, which in turns is \
equal to p2inv under full information. Note that the signal can go to \
infinity, but actual spending is censored. This is computed for the \
equilibrium where people stay uninformed the longest. avginflow is the \
expectation of inflation conditional on being in the region where no \
information acquisition takes place, and avginfhigh is the region in which \
information acquisition takes place. We also compute their (conditional) \
standard deviations\
\>", "Text",
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Cell["\<\
Now, compute equilibrium under endogenous information as a function of the \
posterior based on public information. We select the equilibrium in which \
people stay uninformed the shortest, then we plot both options in the region \
of multiple equilibria. When information is revealed, this coincides with p3, \
so we plot it for several realizations of the shock.\
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Compute value of single deviation in the competitive equilibrium\
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As a first step, I need the value of \[Kappa] in utility terms rather than \
percentage of profits from the experiment above. Profits are u\
\[CloseCurlyQuote]^(-1)(\[Theta]/(\[Theta]-1))/(\[Theta]-1), so I need to \
specify u. \
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Cell["\<\
Next, we construct tables with all the ingredients that go into the utility \
of an informed agent. The outer (first) dimension is the period-3 spending \
and the inner dimension is the period-2 conditional expectation. We need some \
care in the way we extrapolate the utility. The utility of the informed agent \
is decreasing in the beliefs of the other agents up to the point at which the \
period-2 price set by the uninformed agents is equal to the period 3 price, \
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is included  in the region of interpolation, for otherwise the extrapolation \
procedure would miss that the function becomes increasing at some point.\
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The intertemporal elasticity of substitution (denoted as 1/\[Gamma] in what \
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Computation of the value of information at the thresholds that are critical \
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Both vectors represent the value of information at different points when \
shoppers are fully informed. The correct interpretation of the values at \
\[Psi]=0 is that they represent the value of information when a vanishing but \
positive measure of producers become informed, so that the price fully \
reveals information to the shopper. For this reason, valueofinfhighthresh at \
0 does not correspond to the value of information when only a measure zero \
set of producers chooses to acquire information. Both vectors are increasing \
in the posterior mean of spending (it is verified here for only two points, \
but the code can readily be adapted to verify other arbitrary points).

We reach two conclusions:
1. The value of information is globally increasing in the fraction of \
informed producers. Any mixed-strategy equilibrium would be unstable.
2. At criticalbenefitvalue, it is strictly optimal for producers to acquire \
information as soon as even an arbitrarily low fraction becomes informed. \
There is thus a discontinuity in the best response at \[Psi]=0 and there are \
no mixed-strategy equilibria. (Unstable) mixed-strategy equilibria exist \
close to criticalcostvalue, but they stop existing at some value for the \
posterior mean that is strictly between criticalcostvalue and \
criticalbenefitvalue. One may be tempted to conclude that above this \
intermediate threshold the equilibrium where producers choose not to acquire \
information is not robust. However, notice that it is still the case (up to \
criticalbenefitvalue) that producers *strictly* prefer not to acquire \
information when nobody else does. To take a strong stance on this issues it \
would be necessary to smooth the discontinuity by introducing noise traders. \
With noise traders, if only a small fraction of producers acquire \
information, the price would reveal very little, and the value of information \
would remain close to that implied by the case in which nobody acquires \
information, so the equilibrium with no information acquisition would survive \
(up to criticalbenefitvalue)\
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Initialize an array with the optimal price charged by uninformed producers as \
a function of the posterior mean of spending based on public information and \
the fraction of producers that choose to be informed\
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Computation of the value of information at the thresholds that are critical \
for the pure-strategy equilibria: criticalcostvalue (the critical point when \
all other producers choose to acquire information) and criticalbenefitvalue \
(the critical point when all other producers do not choose to acquire \
information). We compute this value for all fractions of producers who may \
choose to acquire information (on the \[Psi]grid).\
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Initialize an array with the optimal price charged by uninformed producers as \
a function of the posterior mean of spending based on public information and \
the fraction of producers that choose to be informed\
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Computation of the value of information at the thresholds that are critical \
for the pure-strategy equilibria: criticalcostvalue (the critical point when \
all other producers choose to acquire information) and criticalbenefitvalue \
(the critical point when all other producers do not choose to acquire \
information). We compute this value for all fractions of producers who may \
choose to acquire information (on the \[Psi]grid).\
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Compute the price index that prevails as a function of spending in the final \
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Computation of the value of information at the thresholds that are critical \
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Compute expected value of information as a function of the signal and \[Psi], \
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Computation of the value of information at the thresholds that are critical \
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Computation of the value of information at the thresholds that are critical \
for the pure-strategy equilibria: criticalcostvalue (the critical point when \
all other producers choose to acquire information) and criticalbenefitvalue \
(the critical point when all other producers do not choose to acquire \
information). We compute this value for all fractions of producers who may \
choose to acquire information (on the \[Psi]grid).\
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Now the value of information is decreasing in \[Psi]. This implies that there \
is no mixed-strategy equilibrium for any value of the posterior mean between \
criticalcostvalue and criticalbenefitvalue, since it would pay off to acquire \
information as soon as shoppers learn the future realization of prices \
(through the choices of any positive measure of producers who choose to \
acquire information). It also implies that there is a range of values below \
criticalcostvalue for which a stable mixed-strategy equilibrium exists, along \
with the equilibrium in which no producer chooses to acquire information. \
Notice that the discontinuity allows the best response to be strict in favor \
of not acquiring information when nobody else does, and yet to be strict in \
favor of acquiring information for values of \[Psi] between 0 and the \
probability implied by the mixed-strategy equilibrium. In practice, we view \
this value of the intertemporal elasticity of substitution as implausibly \
large (and even more so for those that follow). Nonetheless, our results \
about pure-strategy equilibria continue to be correct even for these values.\
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Initialize an array with the optimal price charged by uninformed producers as \
a function of the posterior mean of spending based on public information and \
the fraction of producers that choose to be informed\
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Computation of the value of information at the thresholds that are critical \
for the pure-strategy equilibria: criticalcostvalue (the critical point when \
all other producers choose to acquire information) and criticalbenefitvalue \
(the critical point when all other producers do not choose to acquire \
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Initialize an array with the optimal price charged by uninformed producers as \
a function of the posterior mean of spending based on public information and \
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Computation of the value of information at the thresholds that are critical \
for the pure-strategy equilibria: criticalcostvalue (the critical point when \
all other producers choose to acquire information) and criticalbenefitvalue \
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information). We compute this value for all fractions of producers who may \
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